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Beltrami identity

Let q⁢(t)qtq(t) be a functionℝ→ℝnormal-→ℝℝ\mathbb{R}\to\mathbb{R}, q˙=dd⁢t⁢qnormal-˙qddtq\dot{q}=\frac{d}{d{t}}{q}, and L=L⁢(q,q˙,t)LLqnormal-˙qtL=L(q,\dot{q},t). Begin with the time-relative Euler-Lagrange condition

which is the Beltrami identity. In the calculus of variations, the ability to use the Beltrami identity can vastly simplify problems, and as it happens, many physical problems have ∂∂⁡t⁢L=0tL0\frac{\partial}{\partial{t}}L=0.

In space-relative terms, with q′:=dd⁢x⁢qassignsuperscriptqnormal-′ddxqq^{{\prime}}:=\frac{d}{d{x}}q, we have